Construct a finite menu of experts or LoRA adapters by quantizing the oracle action manifold under a task-aware Fisher-information distortion rather than Euclidean parameter distance. The router can choose an installed expert after observing the input, but only k experts are stored or evaluated. The paper's frontier rate gives a falsifiable accuracy-versus-menu-size prediction.
Insert a fixed or lightly gated lower-triangular binomial-transform layer into a sequence model to create global interactions across positions without forming attention logits. For a sequence of length N, mix each output position with all earlier positions using coefficients determined by an integer m; initialize the layer fixed and optionally learn a diagonal channel gate or a small mixture over m values.
Paper: $m$-Bell and $m$-Stirling numbers: Iterated binomial transforms, hyper-Bessel functions, and moments of the Conway--Maxwell--Poisson distributionarXiv:2608.12011
Tie neural parameters across feature channels according to the Schur multiplicity pattern of the fermionic coinvariant representation. In the two-fermion case, use one learned parameter block for each Schur degree instead of independently parameterizing every ordered pair of fermionic channels, eliminating redundant copies while preserving the relevant GL2 channel symmetry.
Represent the active experts or channels of a sparse layer by a presence set and impose a reaction-style dependency graph on possible support changes. During a growth phase, activate only the least support set closed under enabled dependencies; during later pruning, allow trajectory-dependent removals but never add structurally unreachable experts. This should reduce routing churn and dead experts while preserving adaptive sparsity.